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Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly coNP-complete Pontus Ekberg & Wang Yi Uppsala University ECRTS 2015 Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP

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SLIDE 1

Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly coNP-complete

Pontus Ekberg & Wang Yi

Uppsala University

ECRTS 2015

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SLIDE 2

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-3
SLIDE 3

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-4
SLIDE 4

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-5
SLIDE 5

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-6
SLIDE 6

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-7
SLIDE 7

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-8
SLIDE 8

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-9
SLIDE 9

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-10
SLIDE 10

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-11
SLIDE 11

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-12
SLIDE 12

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

slide-13
SLIDE 13

Context on the Uniprocessor Feasibility Problem

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic)

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-hard

?

Strongly coNP-complete

?

(Weakly) coNP-complete

?

Strongly coNP-complete

?

Pseudo-poly. solution exists

?

1980 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 2

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SLIDE 14

How?

SCP SCP

Strongly NP-complete (Baruah et al., 1990)

∝

Pseudo-polynomial transformation

in-Feasibility in-Feasibility

Strongly NP-hard Strongly coNP-hard

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3

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SLIDE 15

How?

SCP SCP

Strongly NP-complete (Baruah et al., 1990)

∝

Pseudo-polynomial transformation

in-Feasibility in-Feasibility

Strongly NP-hard Strongly coNP-hard

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3

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SLIDE 16

How?

SCP SCP

Strongly NP-complete (Baruah et al., 1990)

∝

Pseudo-polynomial transformation

in-Feasibility in-Feasibility

Strongly NP-hard Strongly coNP-hard

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3

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SLIDE 17

How?

SCP SCP

Strongly NP-complete (Baruah et al., 1990)

∝

Pseudo-polynomial transformation

in-Feasibility in-Feasibility

Strongly NP-hard Strongly coNP-hard

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3

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SLIDE 18

How?

SCP SCP

Strongly NP-complete (Baruah et al., 1990)

∝

Pseudo-polynomial transformation

in-Feasibility in-Feasibility

Strongly NP-hard Strongly coNP-hard

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3

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SLIDE 19

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair A k . Example: A k A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 20

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A k A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 21

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 22

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2 A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 23

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 24

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 25

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 26

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

slide-27
SLIDE 27

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) A ? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 28

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

slide-29
SLIDE 29

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

slide-30
SLIDE 30

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

slide-31
SLIDE 31

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? → Yes A ? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 32

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? → Yes (A, 3)? No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

slide-33
SLIDE 33

The Simultaneous Congruences Problem (SCP)

An SCP instance is given by a pair (A, k). Example: A = {(2, 4), (4, 6), (3, 8), (0, 3)} k = 2

2 4 6 8 10 12 14 16 18 20

(2, 4) (4, 6) (3, 8) (0, 3) (A, 2)? → Yes (A, 3)? → No

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

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SLIDE 34

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

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SLIDE 35

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-36
SLIDE 36

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-37
SLIDE 37

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-38
SLIDE 38

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-39
SLIDE 39

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-40
SLIDE 40

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-41
SLIDE 41

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-42
SLIDE 42

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance A k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-43
SLIDE 43

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance (A, k)

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-44
SLIDE 44

Feasibility and Demand Bound Functions

T = {τ1, τ2, τ3} τ1 τ2 τ3 ∑ Demand bound functions capture feasibility exactly! (Baruah et al., 1990) Feasible / Infeasible SCP instance (A, k)

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 5

slide-45
SLIDE 45

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-46
SLIDE 46

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-47
SLIDE 47

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope =

1 |A|

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-48
SLIDE 48

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-49
SLIDE 49

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-50
SLIDE 50

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-51
SLIDE 51

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope =

1 |A|

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-52
SLIDE 52

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16

(2, 4)

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope =

1 |A|

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-53
SLIDE 53

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-54
SLIDE 54

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k 2 4 6 8 10 12 14 16 2 4

(4, 6)

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-55
SLIDE 55

Constructing the Demand Bound Functions

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

2 4 6 8 10 12 14 16 2 4

(2, 4)

Slope

A

Slope

A

Amount of shifu depends on k 2 4 6 8 10 12 14 16 2 4

(4, 6)

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 6

slide-56
SLIDE 56

Synthesizing the Tasks

A

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-57
SLIDE 57

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)}

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-58
SLIDE 58

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-59
SLIDE 59

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-60
SLIDE 60

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-61
SLIDE 61

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-62
SLIDE 62

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-63
SLIDE 63

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-64
SLIDE 64

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-65
SLIDE 65

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-66
SLIDE 66

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-67
SLIDE 67

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-68
SLIDE 68

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-69
SLIDE 69

Synthesizing the Tasks

A = {(2, 4), (4, 6), (3, 8), (0, 3)} τ1 τ2 τ3 τ4 ∑

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 7

slide-70
SLIDE 70

Conclusion

General case Utilization bounded by a constant c < 1 Asynchronous periodic Synchronous periodic (or sporadic) Strongly coNP-complete Strongly coNP-complete Strongly coNP-complete Pseudo-poly. solution exists

?

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 8

slide-71
SLIDE 71

∀Tiank you! ⋄ ∃Qvestions?

Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 9